Wigner-Seitz unit cell

One has to take into account, however, that the unit cell which is relevant for spectroscopy is the primitive (or Wigner-Seitz) unit cell. It is a parallelepiped from which the entire lattice may be generated by applying multiples of elementary translations. Face- and body-centered cells are multiple unit cells. The content of such a cell has to be divided by a factor m to obtain the content of a primitive unit cell. This factor m is implicitly given by the international symbol for a space group P and R denote primitive cells (m = 1), face-centered cells are denoted A, B, C (m = 2), and F m = 4), and body-centered cells are represented by I m = 2). Examples are described by Turrell (1972). 

Fig. 9.5. Construction of the FBZ as a Wigner-Seitz unit cell of the inverse lattice in 2-D. The circles represent the nodes of the inverse lattice. We cut the lattice in the middle between the origin node W and all the other nodes (here, it turns out to be sufficient to take only the nearest and the next nearest neighbors) and remove all the sawed-off parts that do not contain W. Finally we obtain the FBZ in the form of a hexagon. The Wigner-Seitz unit cells (after performing all allowed translations in the inverse lattice) reproduce the complete inverse space.

As was remarked at the begin-ning of this ehapter, the example of a jigsaw puzzle shows us that a parallelepiped unit eell is not the only ehoice. Now, we win profit from this extra freedom and will define the so-eaUed Wigner-Seitz unit cell. Here is the prescription for... 

A special [Wigner-Seitz) unit cell of the inverse lattice is called the First Brillouin Zone (FBZ). 

The Wigner-Seitz unit cells (after performing all allowed translations in the inverse lattice) reproduce the complete inverse space. 

As was remarked at the beginning of this chapter, the example of a jigsaw puzzle shows us that a parallelepiped unit cell is not the only choice. Now, we will profit from this extra freedom and will define the so-called Wigner-Seitz unit cell. Here is the prescription for how to construct it (Fig. 9.5) We focus on a node W, saw the crystal along the plane that dissects (symmetrically) the distance to a nearest-neighbor node, throw the part that does not contain W into the fireplace, and then repeat the procedure until we are left with a solid containing W. This solid represents the First Brillouin Zone (FBZ). 

All simple induced reps may be generated by induction from the irreps of site-symmetry groups Ggi, of a relatively small number of q points forming the set Q in the Wigner-Seitz unit cell of the direct lattice. The set Q consists of... 

The site groups G, for all q e Q are called maximal isotropy subgroups in [40]. The set Q in the Wigner-Seitz unit cell is determined in the same way as the set K in the BriUouin zone. However, the action of symmetry operations in the direct and reciprocal spaces is different. 

The cycUc cluster can be chosen in the form of the Wigner-Seitz unit cell (shown by the dotted Une in Fig. 6.1 with the center at boron atom 1). A connection of the interaction range to this ceUs atoms was discussed in [301]. 

Also relevant (even though the title name has not been used in this chapter) is The Wigner-Seitz Unit Cell by S. F. A. Kettle and L. J. Norrby, J. Chem. Educ. (1994) 71, 1003. 

One of the earliest numerical solutions of the SCMFT for diblock copolymers was obtained by Helfand and Wasserman [10]. In more recent years, numerical solutions using real-space methods have been obtained by, among others. Vavasour and Whimore [12]. Due to limited computing power, for these early techniques the generally anisotropic Wigner-Seitz unit cells was approximated by a spherical one, so that the SCMFT equations could be reduced to onedimensional form. The spherical-unit-cell approximation has been quite successful in the studies of diblock-copolymer melts [12] and blends [29]. 

Fig. 40 Calculated constant-intensity surface in 4-beam laser interference patterns. The primitive units (contents of Wigner-Seitz unit cell) is shown inset in each case, a Scheme-1, high-index beam vectors interference, producing pattern of 922-nm lattice constant, b Scheme-2, low-index beam vectors interference, producing FCC pattern with 397-nm lattice constant. In both case, the use of 355-nm YAG laser was assumed. Scale bars 500 nm...

The range of k-values between — Ti/a < k < n/siisknownsiSthQ rstBrillouinzone (BZ). The first BZ is also defined as the Wigner-Seitz primitive cell of the reciprocal lattice, whose construction is illustrated in Figure 2.75. First, an arbitrary point in the reciprocal lattice is chosen and vectors are drawn to all nearest-neighbor points. Perpendicular bisector lines are then drawn to each of these vectors the enclosed area corresponds to the primitive unit cell, which is also referred to as the first Brillouin zone. 

Figure 16.3. (a) Construction of the Wigner-Seitz cell in a 2-D hexagonal close-packed (hep) lattice, (b) Primitive unit cell of the hep lattice. 

It is possible, as well, to define the primitive unit cell, by surrounding the lattice points, by planes perpendicularly intersecting the translation vectors between the enclosed lattice point and its nearest neighbors [2,3], In this case, the lattice point will be included in a primitive unit cell type, which is named the Wigner-Seitz cell (see Figure 1.2). 

A band-structure diagram is a map of the variation in the energy, or dispersion, of the extended-wave functions (called bands) for specific Ar-points within the first BZ (also called the Wigner-Seitz cell), which is the unit cell of Ar-space. 

Furthermore, within the (R)APW method the so called muffin-tin approach is used for calculating V(f). According to this model the volume of the unit cell to is separated into the volume tOy of non overlapping and approximately touching atomic spheres (muffin-tin spheres, cf. Fig. 4) centred at the lattice sites y and the volume to between the spheres. In Table 5 the radii ry and the volumes o), which are used for the RAPW calculations of Zintl phases are given. Because of the arrangement of the atoms in the crystal shown in Sect. B, the volumes to a and Wigner-Seitz volumes are listed too. 

From the density, it is now possible to construct the potential for the next iteration, and also to calculate total energies. During the self-consistency loop, we make a spherical approximation to the potential. This is because the computational cost for using the full potential is very high, and that the density converges rather badly in the corners of the unit cell. In order not to confuse matters with the ASA (Atomic Sphere Approximation) this is named the Spherical Cell Approximation (SC A) [65]. If we denote the volume of the Wigner-Seitz cell centered at R by QR, we have fiR = QWR — (4 /3), where w is the Wigner-Seitz radius. This means that the whole space is covered by spheres, just as in the ASA. We will soon see why this is practical, when we try to create the potential. 

The resulting Wigner-Seitz cell of this lattice is a regular hexagon, which can alternatively be viewed as the unit cell. This is illustrated in figure 3. Observe that the symmetry of the replicated system is still cubic. 

The unit cells described above are conventional crystallographic unit cells. However, the method of unit cell construction described is not unique. Other shapes can be found that will fill the space and reproduce the lattice. Although these are not often used in crystallography, they are encountered in other areas of science. The commonest of these is the Wigner-Seitz cell. 

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